| L(s) = 1 | + (−0.00116 + 0.00202i)2-s + (35.8 + 419. i)3-s + (1.02e3 + 1.77e3i)4-s + (−5.47e3 − 9.48e3i)5-s + (−0.889 − 0.416i)6-s + (−3.16e4 + 5.48e4i)7-s − 9.55·8-s + (−1.74e5 + 3.00e4i)9-s + 25.5·10-s + (−2.10e5 + 3.64e5i)11-s + (−7.07e5 + 4.92e5i)12-s + (−4.11e5 − 7.13e5i)13-s + (−73.8 − 127. i)14-s + (3.78e6 − 2.63e6i)15-s + (−2.09e6 + 3.63e6i)16-s + 2.83e6·17-s + ⋯ |
| L(s) = 1 | + (−2.57e−5 + 4.46e−5i)2-s + (0.0851 + 0.996i)3-s + (0.499 + 0.866i)4-s + (−0.784 − 1.35i)5-s + (−4.66e−5 − 2.18e−5i)6-s + (−0.711 + 1.23i)7-s − 0.000103·8-s + (−0.985 + 0.169i)9-s + 8.08e−5·10-s + (−0.393 + 0.681i)11-s + (−0.820 + 0.571i)12-s + (−0.307 − 0.532i)13-s + (−3.67e−5 − 6.35e−5i)14-s + (1.28 − 0.896i)15-s + (−0.499 + 0.866i)16-s + 0.483·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.274919 + 1.01763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.274919 + 1.01763i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-35.8 - 419. i)T \) |
| good | 2 | \( 1 + (0.00116 - 0.00202i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 5 | \( 1 + (5.47e3 + 9.48e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (3.16e4 - 5.48e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (2.10e5 - 3.64e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (4.11e5 + 7.13e5i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 - 2.83e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.56e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-2.24e7 - 3.89e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-2.76e7 + 4.78e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-1.13e8 - 1.95e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 - 2.57e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + (1.99e8 + 3.45e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (2.26e8 - 3.91e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (7.54e8 - 1.30e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 1.27e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (3.21e9 + 5.57e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.21e9 + 5.56e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-6.99e8 - 1.21e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 9.59e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.26e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (2.30e10 - 3.99e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (2.03e9 - 3.52e9i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 - 8.43e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.93e10 + 3.34e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.64964534668149944436037116188, −17.29082925494891560687272069913, −15.90790340109249594627563407628, −15.57641683726916197528921022127, −12.69183555374274310671709198081, −11.78811183425176807060014393774, −9.409504530401470393479452165835, −8.085363929225265073812771125601, −5.10401459998359230577736196083, −3.17929212006584408813129827123,
0.55673875411564008753089460530, 2.98201038740416233042322138112, 6.50328946221734049825595571421, 7.40910325463227646867970571722, 10.37903633640717478995542161451, 11.55255485819817160283816914218, 13.69781116640200215255387837344, 14.76005747768230464530747693248, 16.46637267710132203013672140085, 18.59017277878245550356098105538
